# Avner Ash + Robert Gross

Princeton University Press 2012
A book review by Danny Yee © 2017 http://dannyreviews.com/
Elliptic Tales is a semi-popular introduction to the mathematics of elliptic curves, building up to an explanation of the Birch and Swinnerton-Dyer (BSD) Conjecture, a high profile unsolved problem in mathematics which few have heard about because even stating it requires some moderately complex terminology.
"The conjecture says that the order of the zero of the L-function of an elliptic curve E at s = 1 is equal to the rank of the abelian group E(Q) of rational points on the elliptic curve. In this way, the number of rational solutions to a cubic equation with integer coefficients can be connected to analytic properties of generating functions coming from the solutions of the same equation over all the finite fields."

If most of that is gobbledygook to you, don't give up yet. The goal of Elliptic Tales is to give a mathematically literate but non-specialist reader some feel for what that statement means; it also presents some satisfying intermediate results. Ash and Gross assume only high school algebra and a bit of calculus, explaining what complex numbers are, giving a basic introduction to group theory, and so forth. They do build quite rapidly on that, and the later material gets quite difficult and is probably best suited to university students, but engaged general readers should be able to enjoy part I in detail and skim over part III — which is in any event more cursory, omitting proofs and details and only really outlining the approach to the BSD Conjecture.

Inspired by geometry, Part I investigates how often a line meets an algebraic curve and its connection with the degree of that curve. Introducing the concepts of algebraic closure (we might need to look for solutions in ℂ instead of ℝ, for example), the projective plane (adding a line at infinity so all lines intersect), and intersection multiplicity (to take care of singular points of various kinds), allows the elegant result of Bezout's Theorem. This basically says that the sum of the intersection multiplicities of two projective curves is the product of the degrees of the polynomials defining the curves.

Part II focuses on elliptic curves — nonsingular projective curves defined by homogeneous polynomials of degree three, with integer coefficients — and looks at the number of their rational solutions. It introduces group theory, looking at the group structure on the points of an elliptic curve and analysing this for non-singular and singular cubic equations. Part III introduces further infrastructure: generating functions and various key examples thereof (such as the Riemann-Zeta function), analytic continuation of complex functions, and L-functions. It then recapitulates before stating the BSD Conjecture.

Elliptic Tales seems to me to be more polished than Ash and Gross' earlier Fearless Symmetry, which tackled Fermat's Last Theorem. It is a carefully thought out presentation, starting at a low level, keeping the key ideas and results in focus, decreasing the level of detail as the complexity mounts, and making it possible for the reader to skim the more complex material if necessary while still getting a feel for the broad picture. There are a few exercises, but they are immediately followed by solutions so are more realistically examples.

Its primary audience is probably university mathematics students, perhaps surveying an area of mathematics that is new to them. But anyone with solid high school mathematics and some curiosity should enjoy a good chunk of Elliptic Tales.

February 2017

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